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Category: Limits

Question-153346

Question Number 153346 by mathlove last updated on 06/Sep/21 Answered by liberty last updated on 06/Sep/21 $$\:\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\frac{{f}\left({x}\right)−\mathrm{4}}{{x}−\mathrm{2}}=\mathrm{15}\:\rightarrow\begin{cases}{{f}\left(\mathrm{2}\right)=\mathrm{4}}\\{{f}\:'\left(\mathrm{2}\right)=\mathrm{15}}\end{cases} \\ $$$$\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\frac{\mathrm{2}{x}^{\mathrm{2}} −{f}\left({x}\right)}{{x}−\mathrm{2}}\:=\infty \\ $$ Answered…

f-x-ax-2-bx-1-x-0-cx-2-d-0-lt-x-1-2-bx-d-1-2-lt-x-1-f-x-is-continuous-on-1-1-prove-d-0-c-2b-

Question Number 87799 by M±th+et£s last updated on 06/Apr/20 $${f}\left({x}\right)=\begin{cases}{{ax}^{\mathrm{2}} +{bx}\:\:\:\:\:\:−\mathrm{1}\leqslant{x}\leqslant\mathrm{0}}\\{{cx}^{\mathrm{2}} +{d}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{0}<{x}\leqslant\frac{\mathrm{1}}{\mathrm{2}}}\\{{bx}+{d}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}<{x}\leqslant\mathrm{1}}\end{cases} \\ $$$${f}\left({x}\right)\:{is}\:{continuous}\:{on}\left[−\mathrm{1},\mathrm{1}\right] \\ $$$${prove}\:{d}=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{c}=\mathrm{2}{b} \\ $$ Commented by john santu last…

Question-22165

Question Number 22165 by A1B1C1D1 last updated on 12/Oct/17 Answered by ajfour last updated on 13/Oct/17 $${let}\:\:\frac{{x}}{\mathrm{1}−{x}^{\mathrm{2}} }\:={t}\:\:\:\:\Rightarrow\:{x}^{\mathrm{2}} =\mathrm{1}−\frac{{x}}{{t}} \\ $$$${L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{1}+\frac{{x}}{{t}}−\mathrm{2}\left(\mathrm{1}−\frac{{t}^{\mathrm{2}} }{\mathrm{2}}+\frac{{t}^{\mathrm{4}} }{\mathrm{24}}−….\right)}{\mathrm{2}{x}^{\mathrm{4}} }\right)…

lim-x-0-2sin-x-sin-2x-x-sin-x-

Question Number 87690 by jagoll last updated on 05/Apr/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2sin}\:\mathrm{x}−\mathrm{sin}\:\mathrm{2x}}{\mathrm{x}−\mathrm{sin}\:\mathrm{x}} \\ $$ Commented by jagoll last updated on 05/Apr/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2}\left(\mathrm{x}−\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{6}}\right)−\left(\mathrm{2x}−\frac{\mathrm{8x}^{\mathrm{3}} }{\mathrm{6}}\right)}{\mathrm{x}−\left(\mathrm{x}−\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{6}}\right)}\:=…